x#y pivots y around x in the triangle 1,2,3. Pivot
generates the reflections symmetries of the triple; double reflections are
rotations, so double pivot defines these rotations:

1+x=1#(2#x)=3#(1#x)=2#(3#x) ;

2+x=2#(1#x)=1#(3#x)=3#(2#x).

1+x moves x one step up the cycle 1 < 2 < 3 <
1.

2+x moves x one step down the cycle 1 < 2 < 3
< 1.

Here are function tables for the six permutations of
the triple:

x|123

---------------------|--------------------------------

0
+ x|123

1
+ x|
231

2
+ x|312

1#x|
132

2#x|
321

3#x|213

This is the permutation group on three elements: S3.
It is the same as the symmetries of the triangle. It contains three rotations
(a+x) and three reflections(a#x). Here
is S3’s group table:

a*b | 0+1+2+1#2#3#

_________|_______________________________________

|

0+|0+1+2+1#2#3#

1+|1+2+0+3#1#2#

2+|2+0+1+2#3#1#

1#|1#2#3#0+1+2+

2#|2#3#1#2+0+1+

3#|3#1#2#1+2+0+

Pivot has these laws:

Recall: a # a=a

Commutativity:a # b=b # a

Cancellation:a # (a # b)=b

Level associativity:(a#b) # (c#d)=(a#c) # (b#d)

Transposition:a#b = cif and
only ifa = b#c

Self-distribution:a # (b#c)=(a#b) # (a#c)

If R is a permutation in S3, then R distributes
over #:

S3 symmetry:R(a#b) =Ra # Rb

This is because pivot self-distributes, and every
permutation in S3 derives from pivot.

Pivot operates on triple
ratios thus:

a#(x1, x1,
x3)=(xa#1,
xa#2, xa#3)

For any triple ratios x
and y, and any index a:

a#(x*y)=(a#x)*(a#y)

a#(x/y)=(a#x)/(a#y)

a#1=1

a#0n=0a#n

a#∞n=∞a#n

a#-1n=-1a#n

a#(x+ny)=(a#x)+a#n(a#y)

(1#x)* (2#x)*
(3#x)=1

x * (1+(x))* (2+(x))=1

Since
(a;b;c) corresponds, in the 3 arithmetic, to the dual number ((a/c);(b/c)),
then 3#(a;b;c) corresponds to its conjugate ((b/c);(a/c)) . Likewise, 2#
corresponds, in the 2 arithmetic, to dual conjugation, and 1# is dual
conjugation in the 1 arithmetic.

So the
permutations are conjugations within each of the three arithmetics. But they
also turn the arithmetics into each other.